JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics.1. Introduction. When are two self-adjoint operators unitarily equivalent modulo the trace class? The version of this question in which "trace class" is replaced by "compact" is settled by the Weyl-von Neumann Theorem [1]: If A and B are self-adjoint operators, then there exists a unitary operator U such that UA U* -B is compact if and only if they have the same essential spectrum. Since the essential spectrum of a self-adjoint operator, indeed for any normal operator, coincides with the set of "limit points" of the spectrum (it being understood that any eigenvalue of infinite multiplicity is considered a limit point; [7]) the concept of equivalence modulo the compacts is seen to be a topological property of the spectrum. The question of trace-class equivalence is more delicate and requires a study of additional invariants. This is exemplified by the beautiful theorem of T. Kato [9] and M. Rosenblum [14] which we shall refer to as the Kato-Rosenblum theorem: If A and B are self-adjoint operators for which A -B is trace class, then A and B have unitarily equivalent absolutely continuous parts.In the present note we shall prove the following theorem. THEOREM 1. Two bounded self-adjoint operators A and B are unitarily equivalent modulo the trace class if and only if (1) Aac is unitarily equivalent to Bac (2) aess(A ) = aess(B) (3) there exists a partition of the (countable) sets a (A)\Uess(A) and a(B )\aess(B) a(A)\aess(A) =7Tr(A; B) U 7T,(A; B) a(B )\aess(B) = 7Tr(B;A) U 7T,(B;A) and a bijective map g from T,,, (A;B) to rT,,,(B;A) such that E d(a,fess (A))+ E d(b,Uess(B))+ E la-g(a)I
whe effect of temperature upon ion uptake and respiration was investigated with excised roots ofcorn ( Ze mays) and barley ( Horden_ vgae).A strong Iibition (Qlo -5 to 8) of ion uptake was observed at temperatures below 10 C. At It is well known that low soil temperatures may affect plant growth and that there are great differences among species in their tolerance of low soil temperature (2,(6)(7)(8)13 (approximately 0.07 ,uCi/,umol). This concentration of Rb was assumed to be rate-saturating for the low concentration isotherm mechanism (4). Separate samples of excised roots with a fresh wt of 200 to 300 mg in a "tea bag" (5) were conditioned for 20 min in a 0.5 mM CaSO4 solution at the measurement temperature. The "tea bags" were then transferred into 300 ml of labeled solution at the appropriate temperature for either 20 min below 15 C or 10 min above 15 C.After the uptake period, any 'Rb present in the readily exchangeable "free spaces" of the roots was removed by desorption in 0.05 mM RbNO3 and 0.5 mm CaSO4 at 0 C for I min and 30 min at 25 C. The 'Rb retained by the roots was determined by immersing the roots in 10 ml of liquid scintillation fluid (Toluene, Triton X100, Omnifluor, water, 1000:500:6.66:166) (Fig. 1). The data for ion uptake when graphed as an Arrhenius plot (Fig. 2)
It is shown that the so-called principal function invariant, which is associated in a unitarily invariant way to operators with trace class self commutator TT* -T*T, is invariant under trace class perturbations of T and is an extension of the index of T-z to the whole plane. The connection of the principal function, under additional hypothesis, with the determination of the maximal ideal space of the C* algebra generated by T is discussed, and it is shown that the principal function, even when it takes noninteger values, plays a role in establishing the existence of invariant subspaces for T and in determining the point spectrum of T.This note continues the investigation of the properties of the so-called principal function of an almost normal operator. When T is a bounded completely non-normal operator on a Hilbert space H with self commutator TT* -T*T = 2C/r in trace class, it was shown in a series of papers (1, 2, 4, 10, 13, 14) that a summable function G(5,y), the principal function, could be associated with T in a unitarily invariant way. This function, which we think of as an invariant associated with the C* algebra generated by T, is explicitly and simply computable for many operators. It is known to be a complete unitary invariant for T when C has one dimensional range. It is also invariant under trace class perturbations of T.Several years ago J. D. Pincus conjectured that G(6,y) = Index T -(b + iy) when a + i7 is not in a.ess(T), and established the result, as a completely elementary consequence of classical index results for singular integral operators, provided that strong continuity assumptions were imposed upon the coefficients that enter into a singular integral representation for T.The general conjecture has now been affirmatively settled in two different ways. First J. W. Helton and R. Howe (11) established that a certain representing measure coming from a bilinear functional was a constant multiple of the index times planar Lebesgue measure on the components of the complement of the essential spectrum of T, and J. Pincus (5) showed that the representing measure was actually equal to the principal function times Lebesgue measure. (The precise statements are given in the remarks following Theorems 1 and 4.) Then we showed how the conjecture could be established on the basis of earlier results (4). We give this derivation below.It is known that the map T G is onto (19). To any summable function there corresponds a completely non-normal operator with trace class self commutator and G as principal function. Thus the typical situation is the one in which G rarely if at all assumes integer values.Accordingly, while it is indeed possible and useful to adapt the two-complex variable approach which originally gave rise 1952 to the principal function to the purpose of proving mild index results, by extracting only the relevant bilinear functional and bypassing the rest of the theory, we present results in this paper that seem to require stronger methods. Elementary identitiesThe principal function...
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